In a rock-paper-scissors round robin tournament any two contestants play against each other ten times in a row. Each contestant has a favourite strategy, which is a fixed sequence of ten hands (for example, RRSPPRSPPS), which they play against all other contestants. At the end of the tournament it turned out that every player won at least one hand (out of the ten) against any other player. Prove that at most $1024$ contestants participated in the tournament. [i]Submitted by Dávid Matolcsi, Budapest[/i]

$n\geq 2$ is a given integer. For two permuations $(\alpha_1,\cdots,\alpha_n)$ and $(\beta_1,\cdots,\beta_n)$ of $1,\cdots,n$, consider $n\times n$ matrix $A= \left(a_{ij} \right)_{1\leq i,j\leq n}$ defined by $a_{ij} = (1+\alpha_i \beta_j )^{n-1}$. Find every possible value of $\det(A)$.

Let $ X: \equal{} \{x_1,x_2,\ldots,x_{29}\}$ be a set of $ 29$ boys: they play with each other in a tournament of Pro Evolution Soccer 2009, in respect of the following rules: [list]i) every boy play one and only one time against each other boy (so we can assume that every match has the form $ (x_i \text{ Vs } x_j)$ for some $ i \neq j$); ii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the win of the boy $ x_i$, then $ x_i$ gains $ 1$ point, and $ x_j$ doesn’t gain any point; iii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the parity of the two boys, then $ \frac {1}{2}$ point is assigned to both boys. [/list] (We assume for simplicity that in the imaginary match $ (x_i \text{ Vs } x_i)$ the boy $ x_i$ doesn’t gain any point). Show that for some positive integer $ k \le 29$ there exist a set of boys $ \{x_{t_1},x_{t_2},\ldots,x_{t_k}\} \subseteq X$ such that, for all choice of the positive integer $ i \le 29$, the boy $ x_i$ gains always a integer number of points in the total of the matches $ \{(x_i \text{ Vs } x_{t_1}),(x_i \text{ Vs } x_{t_2}),\ldots, (x_i \text{ Vs } x_{t_k})\}$. [i](Paolo Leonetti)[/i]

[b]a)[/b] Prove that for any natural number $ n\ge 1, $ there is a set $ \mathcal{M} $ of $ n $ points from the Cartesian plane such that the barycenter of every subset of $ \mathcal{M} $ has integral coordinates (both coordinates are integer numbers). [b]b)[/b] Show that if a set $ \mathcal{N} $ formed by an infinite number of points from the Cartesian plane is given such that no three of them are collinear, then there exists a finite subset of $ \mathcal{N} , $ the barycenter of which has non-integral coordinates.

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

Denote by $\mathcal M_2(\mathbb R)$ the set of all $2\times2$ matrices with real entries. Prove that: a) for every $A\in\mathcal M_2(\mathbb R)$ there exist $B,C\in\mathcal M_2(\mathbb R)$ such that $A=B^2+C^2$; b) there do not exist $B,C\in\mathcal M_2(\mathbb R)$ such that $\begin{pmatrix}0&1\\1&0\end{pmatrix}=B^2+C^2$ and $BC=CB$.

In rectangle $ ABCD$, points $ F$ and $ G$ lie on $ \overline{AB}$ so that $ AF \equal{} FG \equal{} GB$ and $ E$ is the midpoint of $ \overline{DC}$. Also, $ \overline{AC}$ intersects $ \overline{EF}$ at $ H$ and $ \overline{EG}$ at $ J$. The area of the rectangle $ ABCD$ is $ 70$. Find the area of triangle $ EHJ$. [asy] size(180); pair A, B, C, D, E, F, G, H, J; A = origin; real length = 6; real width = 3.5; B = length*dir(0); C = (length, width); D = width*dir(90); F = length/3*dir(0); G = 2*length/3*dir(0); E = (length/2, width); H = extension(A, C, E, F); J = extension(A, C, E, G); draw(A--B--C--D--cycle); draw(G--E--F); draw(A--C); label("$A$", A, dir(180)); label("$D$", D, dir(180)); label("$B$", B, dir(0)); label("$C$", C, dir(0)); label("$F$", F, dir(270)); label("$E$", E, dir(90)); label("$G$", G, dir(270)); label("$H$", H, dir(140)); label("$J$", J, dir(340)); [/asy] $ \displaystyle \textbf{(A)} \ \frac {5}{2} \qquad \textbf{(B)} \ \frac {35}{12} \qquad \textbf{(C)} \ 3 \qquad \textbf{(D)} \ \frac {7}{2} \qquad \textbf{(E)} \ \frac {35}{8}$

Let $n \ge 2$ and matrices $A,B \in M_n(\mathbb{R})$. There exist $x \in \mathbb{R} \backslash \{0,\frac{1}{2}, 1 \}$, such that $ xAB + (1-x)BA = I_n$. Show that $(AB-BA)^n = O_n$.

Let $ A,B,C,D $ be four $ 2\times 2 $ complex matrices such that $ A-D $ is invertible and such that $$ A^2+BA+C=0=D^2+BD+C. $$ Prove that $ \text{tr} (A+D) =-\text{tr} B $ and $ \det (AD) =\det C. $

Let $A\in M_4(\mathbb R)$ be an invertible matrix s.t. $\det(A+^tA)=5\det A$ and $\det (A-^tA)=\det A$. Prove that for every complex root $\omega$ of order 5 of unitity (i.e. $\omega^5=1,\omega\not\in\mathbb R$) the following relation holds $\det(\omega A+^tA)=0$. [i]Dan Popescu[/i]

Investigate the convergence of the integral \[\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{dxdy}{1+x^4y^4}\]

Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$

Given a positive integer $n$ and a real valued $n\times n$ matrix $A$. $J$ is $n\times n$ matrix with every entry $1$. Suppose $A$ satisfies the following relations. $$A+A^T = \frac{1}{n} J, \quad AJ = \frac{1}{2} J$$ Show that $A^m-I$ is an invertible matrix for all positive odd integer $m$.

Let $ A\equal{}(a_{ij})_{1\leq i,j\leq n}$ be a real $ n\times n$ matrix, such that $ a_{ij} \plus{} a_{ji} \equal{} 0$, for all $ i,j$. Prove that for all non-negative real numbers $ x,y$ we have \[ \det(A\plus{}xI_n)\cdot \det(A\plus{}yI_n) \geq \det (A\plus{}\sqrt{xy}I_n)^2.\]

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ be two $n \times n$ matrices such that \[ A^2B+BA^2=2ABA \] Prove there exists $k\in \mathbb{N}$ such that \[ (AB-BA)^k=\mathbf{0}_n\] Here $\mathbf{0}_n$ is the null matrix of order $n$.

Let $A\in M_3(\mathbb Z)$ such that $\det(A)=1$. What is the maximum possible number of entries of $A$ that are even?

How many ordered pairs of positive integers $(x,y)$ are there such that $y^2-x^2=2y+7x+4$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{Infinitely many} $

Suppose that $V$ is a finite dimensional vector space over the real numbers equipped with an inner product and $S:V\times V \longrightarrow \mathbb R$ is a skew symmetric function that is linear for each variable when others are kept fixed. Prove there exists a linear transformation $T:V \longrightarrow V$ such that $\forall u,v \in V: S(u,v)=<u,T(v)>$. We know that there always exists $v\in V$ such that $W=<v,T(v)>$ is invariant under $T$. (it means $T(W)\subseteq W$). Prove that if $W$ is invariant under $T$ then the following subspace is also invariant under $T$: $W^{\perp}=\{v\in V:\forall u\in W <v,u>=0\}$. Prove that if dimension of $V$ is more than $3$, then there exist a two dimensional subspace $W$ of $V$ such that the volume defined on it by function $S$ is zero!!!! (This is the way that we can define a two dimensional volume for each subspace $V$. This can be done for volumes of higher dimensions.)

Let $n$ be a positive integer. For any positive integer $k$, let $0_k=diag\{\underbrace{0, ...,0}_{k}\}$ be a $k \times k$ zero matrix. Let $Y=\begin{pmatrix} 0_n & A \\ A^t & 0_{n+1} \end{pmatrix}$ be a $(2n+1) \times (2n+1)$ where $A=(x_{i, j})_{1\leq i \leq n, 1\leq j \leq n+1}$ is a $n \times (n+1)$ real matrix. Let $A^T$ be transpose matrix of $A$ i.e. $(n+1) \times n$ matrix, the element of $(j, i)$ is $x_{i, j}$. (a) Let complex number $\lambda$ be an eigenvalue of $k \times k$ matrix $X$. If there exists nonzero column vectors $v=(x_1, ..., x_k)^t$ such that $Xv=\lambda v$. Prove that 0 is the eigenvalue of $Y$ and the other eigenvalues of $Y$ can be expressed as a form of $\pm \sqrt{\lambda}$ where nonnegative real number $\lambda$ is the eigenvalue of $AA^t$. (b) Let $n=3$ and $a_1$, $a_2$, $a_3$, $a_4$ are $4$ distinct positive real numbers. Let $a=\sqrt[]{\sum_{1\leq i \leq 4}^{}a^{2}_{i}}$ and $x_{i,j}=a_i\delta_{i,j}+a_j\delta_{4,j}-\frac{1}{a^2}(a^2_{i}+a^2_{4})a_j$ where $1\leq i \leq 3, 1\leq j \leq 4$, $\delta_{i, j}= \begin{cases} 1 \text{ if } i=j\\ 0 \text{ if } i\neq j\\ \end{cases}\,$. Prove that $Y$ has 7 distinct eigenvalue.

Determine the number of possible values for the determinant of $A$, given that $A$ is a $n\times n$ matrix with real entries such that $A^3 - A^2 - 3A + 2I = 0$, where $I$ is the identity and $0$ is the all-zero matrix.

In Prime Land, there are seven major cities, labelled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e. we take the indices modulo $7$. Al initially starts at city $C_0$. Each minute for ten minutes, Al flips a fair coin. If the coin land heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\tfrac{m}{1024}$, find $m$. [i]Proposed by Ray Li[/i]

Let $n\in \mathbb{N},\ n\ge 2$. For any matrix $A\in \mathcal{M}_n(\mathbb{C})$, let $m(A)$ be the number of non-zero minors of $A$. Prove that: a)$m(I_n)=2^n-1$; b)If $A\in \mathcal{M}_n(\mathbb{C})$ is non-singular, then $m(A)\ge 2^n-1$. [i]Marius Ghergu[/i]

A real number with absolute value less than $1$ is written in each cell of an $n\times n$ array, so that the sum of the numbers in each $2\times 2$ square is zero. Show that for odd $n$ the sum of all the numbers is less than $n$.

Let $ U$ be an $ n \times n$ orthogonal matrix. Prove that for any $ n \times n$ matrix $ A$, the matrices \[ A_m=\frac{1}{m+1} \sum_{j=0}^m U^{-j}AU^j\] converge entrywise as $ m \rightarrow \infty.$ [i]L. Kovacs[/i]

We consider the following system with $q=2p$: \[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\] in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties: [b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$ [b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$ [b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$